The invention will be explained on the basis of general paraxial calculations performed upon the combination of a corrector, transfer optics and lens in a probe-forming system as applied in STEM (Scanning Transmission Electron Microscopy). Such a system is shown in FIG. 1. It is assumed that the lens is illuminated by an approximately parallel beam, although this is not necessary.
The beam can be regarded as being composed of rays, which propagate parallel to the axis. Each ray is described by
            (                                    r                                                              r              ′                                          )        z    ,whereby, at position z along the optical axis, r is the distance from the ray to the optical axis and r′ is the angle that the ray subtends with the optical axis.
The effect of a (thin) lens without chromatic aberration is described by:
                                          (                                                            r                                                                                                  r                    ′                                                                        )                    out                =                              (                                                            1                                                  0                                                                                                                        -                      1                                        /                    f                                                                    1                                                      )                    ⁢                                    (                                                                    r                                                                                                              r                      ′                                                                                  )                        in                                              [        1        ]            whereby f is the focal length of the lens. Introduction of the effect of first-order axial chromatic aberration in formula [1] gives:
                                          (                                                            r                                                                                                  r                    ′                                                                        )                    out                =                              (                                                            1                                                  0                                                                                                  -                                                                  1                        f                                            ⁡                                              [                                                  1                          -                                                                                                                    C                                c                                                            f                                                        ⁢                                                          Δϕ                                                              ϕ                                lens                                                                                                                                    ]                                                                                                              1                                                      )                    ⁢                                    (                                                                    r                                                                                                              r                      ′                                                                                  )                        in                                              [        2        ]            whereby Cc is the axial chromatic aberration coefficient of the lens, Δφ is the energy spread in the beam, and φlens is the energy of the beam in the lens.
Before giving a more general description of the behavior of a corrector, a simple corrector will first be described, namely a so-called straight-sighted corrector with a telescopic magnification of 1. Such a corrector is depicted in FIG. 1, together with the lens to be corrected.
In the case of a straight-sighted corrector, a ray that is incident parallel to the axis of the corrector (i.e. r′in=0) will also emerge parallel to the axis of the corrector (r′out=0). A magnification of 1 means that this ray will emerge from the corrector at the same height as it entered the corrector (i.e. rout=rin).
For a ray incident upon the corrector parallel to the axis of the corrector, the emergent ray is described by:
                                          (                                                            r                                                                                                  r                    ′                                                                        )                    corr                =                  (                                                                                          r                    0                                    ⁡                                      (                                          1                      +                                              a                        ⁢                                                                                                  ⁢                        Δ                        ⁢                                                                                                  ⁢                                                  ϕ                          /                                                      ϕ                            lens                                                                                                                )                                                                                                                                            -                                          r                      0                                                        ⁢                                      κ                    corr                                    ⁢                  Δ                  ⁢                                                                          ⁢                                      ϕ                    /                                          ϕ                      lens                                                                                                    )                                    [        3        ]            whereby                κcorr indicates the strength of the corrector (whereby κcorr>0); and,        “a” indicates the effect whereby a ray that has a deviant energy will emerge from the detector at a deviant height—the so-called chromatic magnification error.Often, the corrector will be designed in such a manner that a is equal to 0, so that no chromatic magnification error occurs, and also none of the other errors attendant thereto.In order to see now at what height and at what angle the ray is incident upon the lens, the effect of the space between the corrector and the lens—the so-called drift space—is calculated:        
                                                                                          (                                                                                    r                                                                                                                                      r                          ′                                                                                                      )                                                  lens                  ,                  in                                            =                            ⁢                                                (                                                                                    1                                                                    d                                                                                                            0                                                                    1                                                                              )                                ⁢                                  (                                                                                                                                          r                            0                                                    ⁡                                                      (                                                          1                              +                                                              a                                ⁢                                                                                                                                  ⁢                                Δ                                ⁢                                                                                                                                  ⁢                                                                  ϕ                                  /                                                                      ϕ                                    lens                                                                                                                                                        )                                                                                                                                                                                                                    -                                                          r                              0                                                                                ⁢                                                      κ                            corr                                                    ⁢                          Δ                          ⁢                                                                                                          ⁢                                                      ϕ                            /                                                          ϕ                              lens                                                                                                                                                            )                                                                                                        =                            ⁢                              (                                                                                                                              r                          0                                                ⁡                                                  (                                                      1                            +                                                                                          a                                ~                                                            ⁢                                                                                                                          ⁢                              Δ                              ⁢                                                                                                                          ⁢                                                              ϕ                                /                                                                  ϕ                                  lens                                                                                                                                              )                                                                                                                                                                                                  -                                                      r                            0                                                                          ⁢                                                  κ                          corr                                                ⁢                        Δ                        ⁢                                                                                                  ⁢                                                  ϕ                          /                                                      ϕ                            lens                                                                                                                                              )                                                                        [        4        ]            whereby ã=a−d·κcorr and
         (                            1                          d                                      0                          1                      )  is the transfer matrix of the drift space.The image plane of the lens is equal to the rear focal plane of this lens, and thus lies at a distance f behind the lens. The height of the ray in the image plane is obtained by multiplying the vector of the beam that is incident upon the lens by the transfer matrix of the lens given in formula [2] and the transfer matrix of a drift space with a length f.
                                                                                          (                                                                                    r                                                                                                                                      r                          ′                                                                                                      )                                imageplane                            =                            ⁢                                                                    (                                                                                            1                                                                          f                                                                                                                      0                                                                          1                                                                                      )                                    ⁢                                      (                                                                                            1                                                                          0                                                                                                                                                  -                                                                                          1                                f                                                            ⁡                                                              [                                                                  1                                  -                                                                                                                                                    C                                        c                                                                            f                                                                        ⁢                                                                                                                  Δ                                        ⁢                                                                                                                                                                  ⁢                                        ϕ                                                                                                                    ϕ                                        lens                                                                                                                                                                            ]                                                                                                                                                              1                                                                                      )                                    ⁢                                                            (                                                                                                    r                                                                                                                                                              r                              ′                                                                                                                          )                                                              lens                      ,                      in                                                                      =                                                                                        =                            ⁢                                                (                                                                                                                                                                        C                              c                                                        f                                                    ⁢                                                                                    Δ                              ⁢                                                                                                                          ⁢                              ϕ                                                                                      ϕ                              lens                                                                                                                                                  f                                                                                                                                      -                                                                                    1                              f                                                        ⁡                                                          [                                                              1                                -                                                                                                                                            C                                      c                                                                        f                                                                    ⁢                                                                                                            Δ                                      ⁢                                                                                                                                                          ⁢                                      ϕ                                                                                                              ϕ                                      lens                                                                                                                                                                  ]                                                                                                                                                  1                                                                              )                                ⁢                                                      (                                                                                            r                                                                                                                                                  r                            ′                                                                                                                )                                                        lens                    ,                    in                                                                                                          [        5        ]            Substituting equation [4] in equation [5] renders the height of the beam in the image plane of the lens:
                                          (                                                            r                                                                                                  r                    ′                                                                        )                    image                =                              (                                                                                                                              C                        c                                            f                                        ⁢                                                                  Δ                        ⁢                                                                                                  ⁢                        ϕ                                                                    ϕ                        lens                                                                                                              f                                                                                                  -                                                                  1                        f                                            ⁡                                              [                                                  1                          -                                                                                                                    C                                c                                                            f                                                        ⁢                                                                                          Δ                                ⁢                                                                                                                                  ⁢                                ϕ                                                                                            ϕ                                lens                                                                                                                                    ]                                                                                                              1                                                      )                    ⁢                      (                                                                                                      r                      0                                        ⁡                                          (                                              1                        +                                                                              a                            ~                                                    ⁢                                                                                                          ⁢                          Δ                          ⁢                                                                                                          ⁢                                                      ϕ                            /                                                          ϕ                              lens                                                                                                                          )                                                                                                                                                              -                                              r                        0                                                              ⁢                                          κ                      corr                                        ⁢                    Δ                    ⁢                                                                                  ⁢                                          ϕ                      /                                              ϕ                        lens                                                                                                                  )                                              [        6        ]            When terms higher than the first order in Δφ/φlens are neglected, this yields, as regards the height in the image plane of the lens:
                              r          image                =                                            r              0                        ⁡                          (                                                                    C                    c                                    f                                -                                  f                  ⁢                                                                          ⁢                                      κ                    corr                                                              )                                ⁢                                    Δ              ⁢                                                          ⁢              ϕ                                      ϕ              lens                                                          [        7        ]            In the case of a corrected image, this height in the image plane should be equal to 0, and this yields the condition for the strength of the corrector:κcorr=Cc/f2  [8]
A more general elucidation will now be given, again in the case of a straight-sighted corrector with a telescopic magnification of 1, which elucidation can be distinguished in the following ways from the calculations pertaining to the aforementioned, simple corrector:                Transfer optics with a magnification M are located between the corrector and the lens;        These transfer optics, in particular, change the energy of the beam;        The energy of the particles in the corrector is thus unequal to the energy of the particles in the lens;        The effect of instabilities in the corrector power supplies is calculated.The influence of instabilities in the corrector manifests itself in the form of varying powers of the multipoles that form the corrector. As a result of a fluctuation in current (for generating magnetic fields) and voltage (for generating electric fields) the beam will be slightly deflected.As is known to the skilled artisan, a dipole field will give a deflection in one direction, a quadrupole field will give a deflection that is linearly dependent upon the height at which the beam is incident, etc. Accordingly, the influence of instabilities in the power supplies of the corrector, which give rise to additional deflections, can be taken into account in formula [3]:        
                                          (                                                            r                                                                                                  r                    ′                                                                        )                    corr                =                                            T              corr                        ⁡                          (                                                                                          r                      0                                                                                                            0                                                              )                                =                                    (                                                                                          r                      0                                                                                                                                                          -                                                  r                          0                                                                    ⁢                                              κ                        corr                                            ⁢                      Δ                      ⁢                                                                                          ⁢                                              ϕ                        /                                                  ϕ                          corr                                                                                                                                )                        +                          (                                                                    0                                                                                        N                                                              )                                                          [        9        ]            wherein N=N2+N4r0+N6r02+ . . . whereby N2 describes the effect of dipole fields, N4 describes the effect of quadrupole fields, etc.
It ought to be mentioned that the effect of multipole fields is not equally strong in each direction perpendicular to the axis, and that the use of the variable r is, strictly speaking, incorrect. However, seeing as the other elements (lens, drift space) are rotationally symmetric, one can envisage the variable r as being associated with that direction perpendicular to the axis whereby the effect of N is the greatest.
As is known to the skilled artisan, lenses that collectively form the so-called transfer optics are often located between the corrector and the lens. The purpose of these transfer optics is to take the beam that emerges from the corrector and offer it in a suitable manner to the lens. Such a suitable manner may, for example, entail that all particles of a ray that enters the corrector should meet each other in the principal plane of the lens, regardless of their energy. In addition, the transfer optics are characterized by a geometric magnification M.
Seeing as the beam energy φcorr in the corrector differs from the beam energy φlens in the lens to be corrected, acceleration (or deceleration) will occur between the corrector and the lens. This acceleration is assumed to occur in the transfer optics. The transfer matrix of the transfer optics is given by:
                              T          transf                =                  (                                                    M                                            0                                                                    0                                                                                  1                    M                                    ⁢                                                                                    ϕ                        corr                                                                    ϕ                        lens                                                                                                                          )                                    [        10        ]            Naturally, the determinant of the transfer matrix is det(Ttranf)=√{square root over (φcorr/φlens)}, in accordance with the Helmholtz-Lagrange identity.
It ought to be mentioned that the two coefficients that have a value 0 do not have to have this value; however, as is known to the skilled artisan, this is desirable for a parallel-in/parallel-out system whereby further aberrations are small. It is also known to the skilled artisan how the transfer optics can be designed in such a way that the chromatic errors in those transfer optics can be neglected.
After the corrector and transfer optics, one obtains:
                                                        (                                                                    r                                                                                                              r                      ′                                                                                  )                                                      corr                +                            transf                                =                                    (                                                                    M                                                        0                                                                                        0                                                                                                      1                        M                                            ⁢                                                                                                    ϕ                            corr                                                                                ϕ                            lens                                                                                                                                                          )                        ⁢                          (                                                                                          r                      0                                                                                                                                                                                    -                                                      r                            0                                                                          ⁢                                                  κ                          corr                                                ⁢                        Δ                        ⁢                                                                                                  ⁢                                                  ϕ                          /                                                      ϕ                            corr                                                                                              +                      N                                                                                  )                                      ,                            [        11        ]            and in the image plane of the lens, one obtains:
                                                                                          (                                                                                    r                                                                                                                                      r                          ′                                                                                                      )                                image                            =                            ⁢                                                (                                                                                                                                                                        C                              c                                                        f                                                    ⁢                                                                                    Δ                              ⁢                                                                                                                          ⁢                              ϕ                                                                                      ϕ                              lens                                                                                                                                                  f                                                                                                                                      -                                                                                    1                              f                                                        ⁡                                                          [                                                              1                                -                                                                                                                                            C                                      c                                                                        f                                                                    ⁢                                                                                                            Δ                                      ⁢                                                                                                                                                          ⁢                                      ϕ                                                                                                              ϕ                                      lens                                                                                                                                                                  ]                                                                                                                                                  1                                                                              )                                ⁢                                  (                                                                                    M                                                                    0                                                                                                            0                                                                                                                          1                            M                                                    ⁢                                                                                                                    ϕ                                corr                                                                                            ϕ                                lens                                                                                                                                                                                          )                                ⁢                                ⁢                                  (                                                                                                              r                          0                                                                                                                                                                                                                        -                                                              r                                0                                                                                      ⁢                                                          κ                              corr                                                        ⁢                            Δ                            ⁢                                                                                                                  ⁢                                                          ϕ                              /                                                              ϕ                                corr                                                                                                              +                          N                                                                                                      )                                                                                                        =                            ⁢                                                (                                                                                                                                                                        MC                              c                                                        f                                                    ⁢                                                                                    Δ                              ⁢                                                                                                                          ⁢                              ϕ                                                                                      ϕ                              lens                                                                                                                                                                                                        f                            M                                                    ⁢                                                                                                                    ϕ                                corr                                                                                            ϕ                                lens                                                                                                                                                                                                                                                  -                                                                                    M                              f                                                        ⁡                                                          [                                                              1                                -                                                                                                                                            C                                      c                                                                        f                                                                    ⁢                                                                                                            Δ                                      ⁢                                                                                                                                                          ⁢                                      ϕ                                                                                                              ϕ                                      lens                                                                                                                                                                  ]                                                                                                                                                                                                        1                            M                                                    ⁢                                                                                                                    ϕ                                corr                                                                                            ϕ                                lens                                                                                                                                                                                          )                                ⁢                                ⁢                                  (                                                                                                              r                          0                                                                                                                                                                                                                        -                                                              r                                0                                                                                      ⁢                                                          κ                              corr                                                        ⁢                            Δ                            ⁢                                                                                                                  ⁢                                                          ϕ                              /                                                              ϕ                                corr                                                                                                              +                          N                                                                                                      )                                                                                                        =                            ⁢                              (                                                                                                                                                          r                            0                                                    ⁢                                                                                    MC                              c                                                        f                                                    ⁢                                                                                    Δ                              ⁢                                                                                                                          ⁢                              ϕ                                                                                      ϕ                              lens                                                                                                      -                                                                              (                                                                                                                            r                                  0                                                                ⁢                                                                  κ                                  corr                                                                ⁢                                Δ                                ⁢                                                                                                                                  ⁢                                                                  ϕ                                  /                                                                      ϕ                                    corr                                                                                                                              +                              N                                                        )                                                    ⁢                                                      (                                                                                          f                                M                                                            ⁢                                                                                                                                    ϕ                                    corr                                                                                                        ϕ                                    lens                                                                                                                                                        )                                                                                                                                                                                                                                                        -                                                          r                              0                                                                                ⁢                                                                                    M                              f                                                        ⁡                                                          [                                                              1                                -                                                                                                                                            C                                      c                                                                        f                                                                    ⁢                                                                                                            Δ                                      ⁢                                                                                                                                                          ⁢                                      ϕ                                                                                                              ϕ                                      lens                                                                                                                                                                  ]                                                                                                      -                                                                              (                                                                                                                            r                                  0                                                                ⁢                                                                  κ                                  corr                                                                ⁢                                Δ                                ⁢                                                                                                                                  ⁢                                                                  ϕ                                  /                                                                      ϕ                                    corr                                                                                                                              +                              N                                                        )                                                    ⁢                                                      1                            M                                                    ⁢                                                                                                                    ϕ                                corr                                                                                            ϕ                                lens                                                                                                                                                                                                        )                                                                                                                                                                  [                                          12                      a                                        ]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        [                                          12                      b                                        ]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  [                              12                c                            ]                                                                                                                    wherein the first matrix in formula [12a] is the matrix for a thin lens followed by a drift space with length f.The height in the image plane of the lens is thus given by:
                              r          image                =                                            r              0                        ⁢                                          MC                c                            f                        ⁢                                          Δ                ⁢                                                                  ⁢                ϕ                                            ϕ                lens                                              -                                    (                                                                    r                    0                                    ⁢                                      κ                    corr                                    ⁢                  Δ                  ⁢                                                                          ⁢                                      ϕ                    /                                          ϕ                      corr                                                                      +                N                            )                        ⁢                          (                                                f                  M                                ⁢                                                                            ϕ                      corr                                                              ϕ                      lens                                                                                  )                                                          [        13        ]            Eliminating r and substituting α=−Mr0/f, in which α is the opening angle on the image side of the lens, gives the following:
                              r          image                =                                                                                                  -                                          α                      (                                                                        C                          c                                                -                                                                              κ                            corr                                                    ⁢                                                                                    f                              2                                                                                      M                              2                                                                                ⁢                                                                                                                    ϕ                                lens                                                                                            ϕ                                corr                                                                                                                                                        )                                                        ⁢                                                            Δ                      ⁢                                                                                          ⁢                      ϕ                                                              ϕ                      lens                                                                      +                                                                                                          f                  M                                ⁢                                                                            ϕ                      corr                                                              ϕ                      lens                                                                      ⁢                                  (                                                            N                      2                                        -                                                                  N                        4                                            ⁢                      α                      ⁢                                              f                        M                                                                              )                                                                                        [        14        ]            whereby terms N6 and higher have been neglected.Correction of the chromatic aberration of the lens therefore takes place if:
                              κ          corr                =                              C            c                    ⁢                                    M              2                                      f              2                                ⁢                                                                      ϕ                  corr                                                  ϕ                  lens                                                      .                                              [        15        ]            
It ought to be mentioned that
      f    M    ⁢                    ϕ        corr                    ϕ        lens              ⁢      (                  N        2            -                        N          4                ⁢        α        ⁢                  f          M                      )  in formula [14] is hereby assigned a value of 0, seeing as this term gives the effect of instabilities whose magnitude cannot be predicted. Moreover, the effect of this term is different in different directions perpendicular to the axis, as already discussed.
From formula [15], one obtains the magnification M of the transfer optics:
                    M        =                  f          ⁢                                                    κ                corr                                            C                c                                              ⁢                                                    (                                                      ϕ                    lens                                                        ϕ                    corr                                                  )                                            1                /                4                                      .                                              [        16        ]            The position as a result of disturbance fields N in the image plane rimage can be re-written as follows:
                              r          image                =                                            N              2                        ⁢                                                            C                  c                                                  κ                  corr                                                      ⁢                                          (                                                      ϕ                    corr                                                        ϕ                    lens                                                  )                                            3                /                4                                              -                                    N              4                        ⁢            α            ⁢                                          C                c                                            κ                corr                                      ⁢                                                            ϕ                  corr                                                  ϕ                  lens                                            .                                                          [        17        ]            
All of this leads to the insight that, in the case of constant energy φlens of the beam in the lens, constant focal length f and chromatic aberration coefficient Cc of the lens, and constant strength κcorr of the corrector, by choosing φcorr<<φlens, the influence of disturbing deflection fields (=N2) is a factor (φcorr/φlens)3/4 smaller, and (for a constant opening angle α) the effect of disturbing quadrupole fields (=N4) is a factor φcorr/φlens smaller. As a result of this, the demands placed on the stability of the power supplies of the multipole elements are reduced.
An attendant advantage is that, for a constant strength of the corrector, κcorr, the required excitation is smaller. This is because, for a constant excitation (i.e. a constant effect on the particle beam), the currents required to generate magnetic fields are a factor √{square root over (φcorr/φlens)} lower, and the voltages required to generate electric fields are a factor φcorr/φlens lower. As a result of the lower currents and lower voltages, a more compact design of the corrector can be realized.
Another attendant advantage is that the magnification M of the transfer optics is a factor (φlens/φcorr)1/4 larger. For a given opening angle α, the beam diameter in the corrector is consequently smaller, as a result of which undesirable aberrations of the corrector in the image plane become less significant.